Estimation of Value at Risk of a Portfolio

8/7/2019 Estimation of Value at Risk of a Portfolio

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Estimation of Value at Risk of a

Portfolio using Copulas and EVT

Ga ura v Kum a r

2006MT50438

Department of MathematicsIndian Institute of Technology, IIT Delhi

Objectives

Model the returns of each asset.

Find the dependence structure of theunderlying assets. – Model the Joint

distribution. Efficient way to estimate the Value at Risk of a

portfolio consisting of 2-3 assets.

Comparison of results of Copula based model

with the Historical simulation method.

Introduction – Terminology Risk Measurement in a Portfolio is an important concept that is

associated whenever an investment is made in a particular setof stocks. The most popular risk measurement is Value atRisk.

Value at Risk : Defined as the α-quartile from the distribution

function F of the return from a portfolio. The probability thatwe will lose more than V over the next N days is α.

Copulas : In statistics, a copula is used as a general way of formulating a multivariate distribution in such a way thatvarious general types of dependence can be represented.

EVT : Extreme Value Theory : Extreme value theory is a branch of statistics dealing with the extreme deviations fromthe median of probability distributions.

Model The log returns are fed in the Garch model as

G A R C H M O D E L D istrib u tio n M o d e lin gIn n o v a tio n

D e p e n d e n c e

M o d e lin gCopulas

Computing net returns andAdding

α quartile returned as the VaR Backtesting done to Validate the results

Copula generatedreturns

The Portfolio Model Consider the portfolio with nassets. Let r

the log returns of the jth asset at time t.

Each return series is modeled byARMA(1)xGARCH(1,1).

The return series are modeled in a manner thatthe returns are dependent just on theprevious value.

Model is as follows : ε,jt being the random

d istu rb a n ce

−−

t j jt j j jt j

t jt jt j

t jt j j jt jr cr

σ γ ε β α σ η σ ε

An explicit generating mechanism for the Garch(P,Q) for thedisturbance is where η ,jt ,is standardized iid

random draw from some probability distribution.function

α β γ are garch constants and follow certainre strictio n s[ ]1

β j + γj < 1

Where innovations : {η j,t}. J = 1,2,3,…n are noise

processes with mean zero and unit variance

−−

t j jt j j jt j

t jt jt j

t jt j j jt j r cr

σ γ ε β α σ

η σ ε ε φ

t jt jt j ,,,η σ ε =

Fo r a b iv a ria te p o rtfo lio= { , }j 1 2

Modeling marginal distributions Phenomenon of heavy tails. – Handled by EVT The returns of the stocks show high kurtosis –fat

Modeling Innovation Series

Empirical Distribution : The innovation series are fitted in empiricaldistribution to find out the empirical CDF foranalysis. Defined as:

The lower tail losses have not been handledcarefully.

Empirical distribution and GPD Modeling the lower tail (10%) by

Generalized Pareto distribution and the rest byempirical distribution.

Better than the empirical modeling – in Results

Generalized Pareto Distribution (GPD): The GPD is a two parameter distribution with

distribution function - Used for ExtremeValue modeling

G ξ, β (x) = 1-(1+kx/β)-1/k , k< 0 or k > 0

= 1-exp(-x/β) , k =0

> , ≥ W h e re β 0 a n d w h e re x 0> =w h e n k 0

≤ - / < .a n d 0 x βk w h e n k 0

k is th e im p o rta n t sh a p e

p a ra m e te r o f th e d istrib u tio n

w h ile βis a n a d d itio n a l sca lin g.param eter

Courtesy - Wikipedia

Copulas - Application in Finance 2 random variables X and Y with continuous

distribution functions Fx and Fy .

After applying the probability integraltransform:

X1 = Fx

(x) and Y1 = Fy(y) are uniform random

variables. As X and Y are dependent => X1 and Y1 are

dependent. Therefore it’s the same to determine the

dependence structure between X and Y. Theproblem reduces to specify the bivariatedistribution between the 2 uniforms.

That is a copula

Properties of Copulas:A d-dimensional copula C : [0; 1]d [0; 1] is a

function which is a cumulative distributionfunction with uniform marginals.

As cdfs are always increasing, C(u1 ,….., ud) isincreasing in each component ui.

The marginal in component i is obtained bysetting u j = 1 for all j ≠ i and as it must be

uniformly distributed, C(1,….., 1, ui,1,…., 1) = ui

C is bounded

Dependence Modeling - Copulas The Distribution of return of a portfolio depends

on the distribution of individual stocks as wellas on the dependence structure that existsbetween the various stocks in the Portfolio.(modeled effectively by Copulas )

Copulas used in this project:

GumbelCopula, - Stated Directly :Archimedean Family

C(u1,….un)=exp{-[(-logu1)1/ Ѳ

+……+(-logun)1/ Ѳ]Ѳ}

Gaussian Copula, and t-Copula – Derived fromNormal and t-distributions:

Methodology:So far seen : Formation of Return Series

GARCH(1x1) modeling of Return series

Modeling of Innovations (Empirical and Empirical +GPD)

Finding the copula parameter. C(u,v) = uv : is an independent copula For Gumbel copula if Ѳ= 1, C(u,v) = uv;

results in independence.

C(u1,….un)=exp{-[(-logu1)

+……+(-logun)1/ Ѳ]Ѳ}

Estimating VaR for a confidence level α = 10%

Calculate the Backfit ratio – closer to α => better estimate

Data and Implementation : Source : Yahoo Finance. Duration for analysis : 01-01-2005 to 01-

01-2008 Assets in Portfolio:

Bivariate analysis : S&P 500 – Index and Google Inc. Stock Trivariate Analysis : S&P 500, Dow Jones – Index and Google Inc.

Stock.

Implementation:

Matlab– 2009, Toolboxes used – Econometrictoolbox.

Simulation run on – 4GB , 2.26 Ghz.

Results and Discussions:

G o o g le

tatistics on log returns &P 500 .oo gl e I nc

Mean . -2 6573e 004 .0 0016

tandard Deviation .0 0078 .0 0192

,a ximu m Mi ni mum . , - .0 0288 0 0353 . , - .0 1143 0 0886

x cess o f K ur tos is .5 1675 .6 4646

:Statistics on returns

: & .Table 1 Statistics of the log returns of S P 500 and Google Inc respectively

Innovations Scatter Plot: Clearly visible returns are related – positive

corelation between the returns.

Simulate 1000 times and average out tofind the correct VaR estimate. Once the innovation series are obtained these are

then modeled using Copulas to find thedependence structure between the underlyingstocks of the portfolio.

The net returns are then calculated by fitting in theparameters of the GARCH and thereby finding anestimated log return series.

We convert these estimated log return series to thenormal return series and calculate the 10%quartile as the VaR.

Copula Parameter

umbel Copula . ( )0 6566 1 implies independence

Comparisons of Models The simulation is also run for to calculate the

VaR obtained by historical simulation method.Here we do not consider any dependencestructure between them

Still does not convince if the VaR estimated isunderestimated or overestimated.

Therefore we apply a back fit test.

Method ( %)aR 10

–umbel Copula based VaR .10 3502

istorical Simulation .5 8400

Back Fit Test Once the VaR is obtained we can obtain the

back fit ratio to check the accuracy of the VaRobtained by different methods.(Number of observations below VaR obtained)

Method

–umbel Copula based VaR / = . –81 754 0 107 Clearly shows that its better

istorical Simulation / = .39 754 0 0517

Comparison of Innovation ModelingEmpirical vs. Empirical GPD

Therefore concludes that the modeling is better andaccurate results are obtained when the lower taillosses are taken into consideration

-umbel Copula m pi ri ca l a nd G PD d is tr ib ut io n mpirical Distribution

= . ( %)aR at α 0 1 10 .0 3 50 2 .9 5360

a ck fi t R at io .10 7 .0 091

opula Parameter . ( –> )6566 Lower more dependence .0 6912

Extending to 3 stocks:

Results – t-Copula

&P 500 GOOGLE DOW

&P .1 0000 .0 4067 .0 9680

GOOGLE.

0 4067.

1 0000,

0 3594

DOW .0 9680 .0 3594 .1 0000

: -Table Co relation Matrix obtained by

.copula fitting

-Dependence structure modeled by t copula

-Copula with Empirical and GPDdistribution

= . ( %)aR at α 0 1 10 .6 6 524

ackfit Ratio .09 3

:Table VaR and Backfit.test and comparison

Future Work Working to find the goodness of fit of copulas

for the estimation of VaR. This could besimulated by measuring the distance of thecopula (Gaussian, Gumbel or the t-copula)with the empirical copula and then comment

on the fitness of the copula used. Some more comparisons :Comparisons with various other techniques of

VaRestimation: Few to name : EWMA model,BeKKModel, Egarch, Tgarch and Montecarlosimulation.

Current research on misspecified marginals andcopulas is rather rare. Thereby would be achallenge to check the effects on theestimation if the marginals are not specifiedcorrectly. This can be modeled using a

montecarlo simulation as suggested byDeanFantazzini in his paper of 2008-09.

Brief study on the Limitations of Copulas :Would also study the downfall of financial

markets in 2008.

References:

Nelson R. B. (1999), An Introduction to Copulas, Springer Verlag Palaro P.H. and HottaL. K. - Using conditional copula to Estimate

Value at Risk, Journal of Data Science 4.(2006) pg 93-115 Genest C. and Anne-Catherine Favre – Everything You Always

Wanted to Know about Copula Modeling but Were Afraid to Ask – Journal of Hydrologic Engineering,Vol.12,No.4

Fantazzini D. - The effects of misspecified marginals and copulas oncomputing the value at risk : A Monte Carlo study ComputationalStatistics and Data Analysis 53(2009)2168–2188

M. Dorey , P. Joubert - Modelling Copulas: An Overview – Article inActuarial society

Quesada-Molina J.J , Rodrguez-LallenaJ. A., Ubeda-Flores M. - Whatare copulas ? Monografas del Semin. Matem.Garcıade Galdea no.

27: 499–506,(2003) Mikosch T.- Copulas: Tales and Facts, 4th International Conference onExtreme Value Analysis in Gothenburg,

http:www.portfolio.com/business-news/2009/03/03/Formula-That-Killed-Wall-Street

Thank You !

Estimation of Value at Risk of a Portfolio

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